*DECK CAIRY
SUBROUTINE CAIRY (Z, ID, KODE, AI, NZ, IERR)
C***BEGIN PROLOGUE CAIRY
C***PURPOSE Compute the Airy function Ai(z) or its derivative dAi/dz
C for complex argument z. A scaling option is available
C to help avoid underflow and overflow.
C***LIBRARY SLATEC
C***CATEGORY C10D
C***TYPE COMPLEX (CAIRY-C, ZAIRY-C)
C***KEYWORDS AIRY FUNCTION, BESSEL FUNCTION OF ORDER ONE THIRD,
C BESSEL FUNCTION OF ORDER TWO THIRDS
C***AUTHOR Amos, D. E., (SNL)
C***DESCRIPTION
C
C On KODE=1, CAIRY computes the complex Airy function Ai(z)
C or its derivative dAi/dz on ID=0 or ID=1 respectively. On
C KODE=2, a scaling option exp(zeta)*Ai(z) or exp(zeta)*dAi/dz
C is provided to remove the exponential decay in -pi/31 and from power series when abs(z)<=1.
C
C In most complex variable computation, one must evaluate ele-
C mentary functions. When the magnitude of Z is large, losses
C of significance by argument reduction occur. Consequently, if
C the magnitude of ZETA=(2/3)*Z**(3/2) exceeds U1=SQRT(0.5/UR),
C then losses exceeding half precision are likely and an error
C flag IERR=3 is triggered where UR=R1MACH(4)=UNIT ROUNDOFF.
C Also, if the magnitude of ZETA is larger than U2=0.5/UR, then
C all significance is lost and IERR=4. In order to use the INT
C function, ZETA must be further restricted not to exceed
C U3=I1MACH(9)=LARGEST INTEGER. Thus, the magnitude of ZETA
C must be restricted by MIN(U2,U3). In IEEE arithmetic, U1,U2,
C and U3 are approximately 2.0E+3, 4.2E+6, 2.1E+9 in single
C precision and 4.7E+7, 2.3E+15, 2.1E+9 in double precision.
C This makes U2 limiting is single precision and U3 limiting
C in double precision. This means that the magnitude of Z
C cannot exceed approximately 3.4E+4 in single precision and
C 2.1E+6 in double precision. This also means that one can
C expect to retain, in the worst cases on 32-bit machines,
C no digits in single precision and only 6 digits in double
C precision.
C
C The approximate relative error in the magnitude of a complex
C Bessel function can be expressed as P*10**S where P=MAX(UNIT
C ROUNDOFF,1.0E-18) is the nominal precision and 10**S repre-
C sents the increase in error due to argument reduction in the
C elementary functions. Here, S=MAX(1,ABS(LOG10(ABS(Z))),
C ABS(LOG10(FNU))) approximately (i.e., S=MAX(1,ABS(EXPONENT OF
C ABS(Z),ABS(EXPONENT OF FNU)) ). However, the phase angle may
C have only absolute accuracy. This is most likely to occur
C when one component (in magnitude) is larger than the other by
C several orders of magnitude. If one component is 10**K larger
C than the other, then one can expect only MAX(ABS(LOG10(P))-K,
C 0) significant digits; or, stated another way, when K exceeds
C the exponent of P, no significant digits remain in the smaller
C component. However, the phase angle retains absolute accuracy
C because, in complex arithmetic with precision P, the smaller
C component will not (as a rule) decrease below P times the
C magnitude of the larger component. In these extreme cases,
C the principal phase angle is on the order of +P, -P, PI/2-P,
C or -PI/2+P.
C
C***REFERENCES 1. M. Abramowitz and I. A. Stegun, Handbook of Mathe-
C matical Functions, National Bureau of Standards
C Applied Mathematics Series 55, U. S. Department
C of Commerce, Tenth Printing (1972) or later.
C 2. D. E. Amos, Computation of Bessel Functions of
C Complex Argument and Large Order, Report SAND83-0643,
C Sandia National Laboratories, Albuquerque, NM, May
C 1983.
C 3. D. E. Amos, A Subroutine Package for Bessel Functions
C of a Complex Argument and Nonnegative Order, Report
C SAND85-1018, Sandia National Laboratory, Albuquerque,
C NM, May 1985.
C 4. D. E. Amos, A portable package for Bessel functions
C of a complex argument and nonnegative order, ACM
C Transactions on Mathematical Software, 12 (September
C 1986), pp. 265-273.
C
C***ROUTINES CALLED CACAI, CBKNU, I1MACH, R1MACH
C***REVISION HISTORY (YYMMDD)
C 830501 DATE WRITTEN
C 890801 REVISION DATE from Version 3.2
C 910415 Prologue converted to Version 4.0 format. (BAB)
C 920128 Category corrected. (WRB)
C 920811 Prologue revised. (DWL)
C***END PROLOGUE CAIRY
COMPLEX AI, CONE, CSQ, CY, S1, S2, TRM1, TRM2, Z, ZTA, Z3
REAL AA, AD, AK, ALIM, ATRM, AZ, AZ3, BK, CK, COEF, C1, C2, DIG,
* DK, D1, D2, ELIM, FID, FNU, RL, R1M5, SFAC, TOL, TTH, ZI, ZR,
* Z3I, Z3R, R1MACH, BB, ALAZ
INTEGER ID, IERR, IFLAG, K, KODE, K1, K2, MR, NN, NZ, I1MACH
DIMENSION CY(1)
DATA TTH, C1, C2, COEF /6.66666666666666667E-01,
* 3.55028053887817240E-01,2.58819403792806799E-01,
* 1.83776298473930683E-01/
DATA CONE / (1.0E0,0.0E0) /
C***FIRST EXECUTABLE STATEMENT CAIRY
IERR = 0
NZ=0
IF (ID.LT.0 .OR. ID.GT.1) IERR=1
IF (KODE.LT.1 .OR. KODE.GT.2) IERR=1
IF (IERR.NE.0) RETURN
AZ = ABS(Z)
TOL = MAX(R1MACH(4),1.0E-18)
FID = ID
IF (AZ.GT.1.0E0) GO TO 60
C-----------------------------------------------------------------------
C POWER SERIES FOR ABS(Z).LE.1.
C-----------------------------------------------------------------------
S1 = CONE
S2 = CONE
IF (AZ.LT.TOL) GO TO 160
AA = AZ*AZ
IF (AA.LT.TOL/AZ) GO TO 40
TRM1 = CONE
TRM2 = CONE
ATRM = 1.0E0
Z3 = Z*Z*Z
AZ3 = AZ*AA
AK = 2.0E0 + FID
BK = 3.0E0 - FID - FID
CK = 4.0E0 - FID
DK = 3.0E0 + FID + FID
D1 = AK*DK
D2 = BK*CK
AD = MIN(D1,D2)
AK = 24.0E0 + 9.0E0*FID
BK = 30.0E0 - 9.0E0*FID
Z3R = REAL(Z3)
Z3I = AIMAG(Z3)
DO 30 K=1,25
TRM1 = TRM1*CMPLX(Z3R/D1,Z3I/D1)
S1 = S1 + TRM1
TRM2 = TRM2*CMPLX(Z3R/D2,Z3I/D2)
S2 = S2 + TRM2
ATRM = ATRM*AZ3/AD
D1 = D1 + AK
D2 = D2 + BK
AD = MIN(D1,D2)
IF (ATRM.LT.TOL*AD) GO TO 40
AK = AK + 18.0E0
BK = BK + 18.0E0
30 CONTINUE
40 CONTINUE
IF (ID.EQ.1) GO TO 50
AI = S1*CMPLX(C1,0.0E0) - Z*S2*CMPLX(C2,0.0E0)
IF (KODE.EQ.1) RETURN
ZTA = Z*CSQRT(Z)*CMPLX(TTH,0.0E0)
AI = AI*CEXP(ZTA)
RETURN
50 CONTINUE
AI = -S2*CMPLX(C2,0.0E0)
IF (AZ.GT.TOL) AI = AI + Z*Z*S1*CMPLX(C1/(1.0E0+FID),0.0E0)
IF (KODE.EQ.1) RETURN
ZTA = Z*CSQRT(Z)*CMPLX(TTH,0.0E0)
AI = AI*CEXP(ZTA)
RETURN
C-----------------------------------------------------------------------
C CASE FOR ABS(Z).GT.1.0
C-----------------------------------------------------------------------
60 CONTINUE
FNU = (1.0E0+FID)/3.0E0
C-----------------------------------------------------------------------
C SET PARAMETERS RELATED TO MACHINE CONSTANTS.
C TOL IS THE APPROXIMATE UNIT ROUNDOFF LIMITED TO 1.0E-18.
C ELIM IS THE APPROXIMATE EXPONENTIAL OVER- AND UNDERFLOW LIMIT.
C EXP(-ELIM).LT.EXP(-ALIM)=EXP(-ELIM)/TOL AND
C EXP(ELIM).GT.EXP(ALIM)=EXP(ELIM)*TOL ARE INTERVALS NEAR
C UNDERFLOW AND OVERFLOW LIMITS WHERE SCALED ARITHMETIC IS DONE.
C RL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC EXPANSION FOR LARGE Z.
C DIG = NUMBER OF BASE 10 DIGITS IN TOL = 10**(-DIG).
C-----------------------------------------------------------------------
K1 = I1MACH(12)
K2 = I1MACH(13)
R1M5 = R1MACH(5)
K = MIN(ABS(K1),ABS(K2))
ELIM = 2.303E0*(K*R1M5-3.0E0)
K1 = I1MACH(11) - 1
AA = R1M5*K1
DIG = MIN(AA,18.0E0)
AA = AA*2.303E0
ALIM = ELIM + MAX(-AA,-41.45E0)
RL = 1.2E0*DIG + 3.0E0
ALAZ=ALOG(AZ)
C-----------------------------------------------------------------------
C TEST FOR RANGE
C-----------------------------------------------------------------------
AA=0.5E0/TOL
BB=I1MACH(9)*0.5E0
AA=MIN(AA,BB)
AA=AA**TTH
IF (AZ.GT.AA) GO TO 260
AA=SQRT(AA)
IF (AZ.GT.AA) IERR=3
CSQ=CSQRT(Z)
ZTA=Z*CSQ*CMPLX(TTH,0.0E0)
C-----------------------------------------------------------------------
C RE(ZTA).LE.0 WHEN RE(Z).LT.0, ESPECIALLY WHEN IM(Z) IS SMALL
C-----------------------------------------------------------------------
IFLAG = 0
SFAC = 1.0E0
ZI = AIMAG(Z)
ZR = REAL(Z)
AK = AIMAG(ZTA)
IF (ZR.GE.0.0E0) GO TO 70
BK = REAL(ZTA)
CK = -ABS(BK)
ZTA = CMPLX(CK,AK)
70 CONTINUE
IF (ZI.NE.0.0E0) GO TO 80
IF (ZR.GT.0.0E0) GO TO 80
ZTA = CMPLX(0.0E0,AK)
80 CONTINUE
AA = REAL(ZTA)
IF (AA.GE.0.0E0 .AND. ZR.GT.0.0E0) GO TO 100
IF (KODE.EQ.2) GO TO 90
C-----------------------------------------------------------------------
C OVERFLOW TEST
C-----------------------------------------------------------------------
IF (AA.GT.(-ALIM)) GO TO 90
AA = -AA + 0.25E0*ALAZ
IFLAG = 1
SFAC = TOL
IF (AA.GT.ELIM) GO TO 240
90 CONTINUE
C-----------------------------------------------------------------------
C CBKNU AND CACAI RETURN EXP(ZTA)*K(FNU,ZTA) ON KODE=2
C-----------------------------------------------------------------------
MR = 1
IF (ZI.LT.0.0E0) MR = -1
CALL CACAI(ZTA, FNU, KODE, MR, 1, CY, NN, RL, TOL, ELIM, ALIM)
IF (NN.LT.0) GO TO 250
NZ = NZ + NN
GO TO 120
100 CONTINUE
IF (KODE.EQ.2) GO TO 110
C-----------------------------------------------------------------------
C UNDERFLOW TEST
C-----------------------------------------------------------------------
IF (AA.LT.ALIM) GO TO 110
AA = -AA - 0.25E0*ALAZ
IFLAG = 2
SFAC = 1.0E0/TOL
IF (AA.LT.(-ELIM)) GO TO 180
110 CONTINUE
CALL CBKNU(ZTA, FNU, KODE, 1, CY, NZ, TOL, ELIM, ALIM)
120 CONTINUE
S1 = CY(1)*CMPLX(COEF,0.0E0)
IF (IFLAG.NE.0) GO TO 140
IF (ID.EQ.1) GO TO 130
AI = CSQ*S1
RETURN
130 AI = -Z*S1
RETURN
140 CONTINUE
S1 = S1*CMPLX(SFAC,0.0E0)
IF (ID.EQ.1) GO TO 150
S1 = S1*CSQ
AI = S1*CMPLX(1.0E0/SFAC,0.0E0)
RETURN
150 CONTINUE
S1 = -S1*Z
AI = S1*CMPLX(1.0E0/SFAC,0.0E0)
RETURN
160 CONTINUE
AA = 1.0E+3*R1MACH(1)
S1 = CMPLX(0.0E0,0.0E0)
IF (ID.EQ.1) GO TO 170
IF (AZ.GT.AA) S1 = CMPLX(C2,0.0E0)*Z
AI = CMPLX(C1,0.0E0) - S1
RETURN
170 CONTINUE
AI = -CMPLX(C2,0.0E0)
AA = SQRT(AA)
IF (AZ.GT.AA) S1 = Z*Z*CMPLX(0.5E0,0.0E0)
AI = AI + S1*CMPLX(C1,0.0E0)
RETURN
180 CONTINUE
NZ = 1
AI = CMPLX(0.0E0,0.0E0)
RETURN
240 CONTINUE
NZ = 0
IERR=2
RETURN
250 CONTINUE
IF(NN.EQ.(-1)) GO TO 240
NZ=0
IERR=5
RETURN
260 CONTINUE
IERR=4
NZ=0
RETURN
END